What is local consistency in constraint satisfaction problems?
If we can do some reduction in the search space using CSP (constraint propagation) we can drastically reduce the search space or sometimes completely avoid the need for a search by directly reaching the solutions (for e.g. with variables having their domains reduced to size one). It could also happen that we come to a point when a variable domain size becomes zero, in that case no solution exists, given the constraints, so no need for a search. 

Constraint propagation basically involves the concept of enforcing local consistency (this is done by enforcing node-consistency, arc-consistency, path-consistency and also global constraints using Alldiif or Atmost). 

The terms: nodes, arc, path, etc. basically reflects a CSP problem represented as a graph with nodes as the variables and the arcs/edges as constraints. The process is simply to remove values from the domains of the variables that are inconsistent. Algorithms such as AC-3, PC-2, etc. precisely are for these purposes.